principled models in the analysis of social relationships

the case of agonistic dominance

Christof Neumann
christofneumann1@gmail.com

Cognitive Ethology Lab, German Primate Center

defining ad hockery

the practice of reacting to what happens or is needed at a particular time, rather than planning in a way that suits all possible situations

  • lack of formal grounding

  • coming up with something just like that

    • etwas aus dem Ärmel schütteln

    • faire qc. en un simple claquement de doigts

ad hockery is everywhere common

  • we should ground things in substantial theory

  • or at least try to get more precise definitions and operationalizations

  • what is steepness, exactly?

  • what is degree of female dominance, exactly?

  • what is bond strength, exactly?

what can we do?

  • in a nutshell we need to parameterize things = put our analyses on a solid basis

  • parameter: unknown quantity that governs the behavior of a probability distribution or a causal mechanism

  • focus today: parameterize features of sociality (dominance)

  • ideally, these features are tractable (no AI here)

  • with parameters:

    • we can criticize our models, improve them, use them as pieces in joint models

    • we can reverse-engineer -> simulate the data generating process

analysis of dominance interactions

  • iterative algorithms over the entire interaction network (e.g. I&SI, de Vries (1998), ADAGIO, Douglas, Ngonga Ngomo, and Hohmann (2017))

  • based on dyadic proportions (e.g. DS, David (1987), CBI, Clutton-Brock et al. (1979))

  • sequential analysis (e.g. Elo-rating, Elo (1978), Glicko, Glickman (1999))

  • ‘exotics’ (e.g. Fushing, McAssey, and McCowan (2011), Adams (2005))

workflow

  • interaction data (either matrix or sequence): a subset of true interactions
  • individual-level
    • dominance hierarchy with ordinal ranks
    • and/or dominance scores/ratings
  • group-level
    • linearity, transitivity
    • steepness
    • degree of female dominance

interaction data you say?

Date Time winner loser Draw
2000-01-01 12:56 x r FALSE
2000-01-02 17:03 v o FALSE
2000-01-03 11:38 r o FALSE
2000-01-04 13:00 r v FALSE
2000-01-05 12:33 q v FALSE
2000-01-06 16:22 o v FALSE
2000-01-07 06:42 o v FALSE
2000-01-08 09:13 o q FALSE
2000-01-09 13:17 q x FALSE
2000-01-10 14:58 q r FALSE
2000-01-11 14:50 o q FALSE
2000-01-12 12:32 r x FALSE
2000-01-13 09:20 r x FALSE
2000-01-14 12:20 o r FALSE
2000-01-15 17:41 v r FALSE
2000-01-16 11:53 o q FALSE
2000-01-17 17:45 q v FALSE
2000-01-18 17:09 q r FALSE
2000-01-19 06:06 v o FALSE
2000-01-20 10:08 o v FALSE
2000-01-21 11:14 r x FALSE
2000-01-22 06:20 o x FALSE
2000-01-23 14:36 v q FALSE
2000-01-24 12:40 q v FALSE
2000-01-25 12:24 v x FALSE
2000-01-26 08:45 q o FALSE
2000-01-27 11:36 r v FALSE
2000-01-28 07:36 q o FALSE
2000-01-29 15:33 o v FALSE
2000-01-30 16:41 o x FALSE
2000-01-31 15:24 r v FALSE
2000-02-01 12:43 v x FALSE
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-05 15:34 o x FALSE
2000-02-06 06:05 v q FALSE
2000-02-07 16:09 q o FALSE
2000-02-08 16:41 o x FALSE
2000-02-09 15:37 q v FALSE
2000-02-10 15:46 q x FALSE
2000-02-11 13:19 q v FALSE
2000-02-12 15:27 x o FALSE
2000-02-13 10:19 v x FALSE
2000-02-14 17:57 q o FALSE
2000-02-15 12:43 o v FALSE
2000-02-16 13:56 o r FALSE
2000-02-17 10:22 o r FALSE
2000-02-18 17:05 o q FALSE
2000-02-19 11:56 r v FALSE
2000-02-20 13:39 r x FALSE
2000-02-21 06:43 o x FALSE
2000-02-22 08:24 o r FALSE
2000-02-23 15:54 o r FALSE
2000-02-24 08:55 x v FALSE
2000-02-25 16:05 r v FALSE
2000-02-26 06:57 v x FALSE
2000-02-27 11:38 o v FALSE
2000-02-28 11:50 q v FALSE
2000-02-29 17:41 o v FALSE
2000-03-01 09:05 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-03 10:31 q x FALSE
2000-03-04 06:13 o x FALSE
2000-03-05 08:01 v x FALSE
2000-03-06 11:44 o x FALSE
2000-03-07 08:17 o v FALSE
2000-03-08 12:21 v x FALSE
2000-03-09 08:13 o r FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE
2000-03-12 06:50 q r FALSE
2000-03-13 14:50 r x FALSE
2000-03-14 17:39 q v FALSE
2000-03-15 07:05 v o FALSE
Date Time winner loser Draw
2000-01-01 12:56 x r FALSE
2000-01-02 17:03 v o FALSE
2000-01-03 11:38 r o FALSE
2000-01-04 13:00 r v FALSE
2000-01-05 12:33 q v FALSE
2000-01-06 16:22 o v FALSE
2000-01-07 06:42 o v FALSE
2000-01-08 09:13 o q FALSE
2000-01-09 13:17 q x FALSE
2000-01-10 14:58 q r FALSE
2000-01-11 14:50 o q FALSE
2000-01-12 12:32 r x FALSE
2000-01-13 09:20 r x FALSE
2000-01-14 12:20 o r FALSE
2000-01-15 17:41 v r FALSE
2000-01-16 11:53 o q FALSE
2000-01-17 17:45 q v FALSE
2000-01-18 17:09 q r FALSE
2000-01-19 06:06 v o FALSE
2000-01-20 10:08 o v FALSE
2000-01-21 11:14 r x FALSE
2000-01-22 06:20 o x FALSE
2000-01-23 14:36 v q FALSE
2000-01-24 12:40 q v FALSE
2000-01-25 12:24 v x FALSE
2000-01-26 08:45 q o FALSE
2000-01-27 11:36 r v FALSE
2000-01-28 07:36 q o FALSE
2000-01-29 15:33 o v FALSE
2000-01-30 16:41 o x FALSE
2000-01-31 15:24 r v FALSE
2000-02-01 12:43 v x FALSE
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-05 15:34 o x FALSE
2000-02-06 06:05 v q FALSE
2000-02-07 16:09 q o FALSE
2000-02-08 16:41 o x FALSE
2000-02-09 15:37 q v FALSE
2000-02-10 15:46 q x FALSE
2000-02-11 13:19 q v FALSE
2000-02-12 15:27 x o FALSE
2000-02-13 10:19 v x FALSE
2000-02-14 17:57 q o FALSE
2000-02-15 12:43 o v FALSE
2000-02-16 13:56 o r FALSE
2000-02-17 10:22 o r FALSE
2000-02-18 17:05 o q FALSE
2000-02-19 11:56 r v FALSE
2000-02-20 13:39 r x FALSE
2000-02-21 06:43 o x FALSE
2000-02-22 08:24 o r FALSE
2000-02-23 15:54 o r FALSE
2000-02-24 08:55 x v FALSE
2000-02-25 16:05 r v FALSE
2000-02-26 06:57 v x FALSE
2000-02-27 11:38 o v FALSE
2000-02-28 11:50 q v FALSE
2000-02-29 17:41 o v FALSE
2000-03-01 09:05 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-03 10:31 q x FALSE
2000-03-04 06:13 o x FALSE
2000-03-05 08:01 v x FALSE
2000-03-06 11:44 o x FALSE
2000-03-07 08:17 o v FALSE
2000-03-08 12:21 v x FALSE
2000-03-09 08:13 o r FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE
2000-03-12 06:50 q r FALSE
2000-03-13 14:50 r x FALSE
2000-03-14 17:39 q v FALSE
2000-03-15 07:05 v o FALSE
Date Time winner loser Draw
2000-02-02 17:14 v o FALSE
2000-02-03 08:32 o x FALSE
2000-02-04 07:19 v x FALSE
2000-02-08 16:41 o x FALSE
2000-02-18 17:05 o q FALSE
2000-02-21 06:43 o x FALSE
2000-02-26 06:57 v x FALSE
2000-02-29 17:41 o v FALSE
2000-03-02 14:23 q v FALSE
2000-03-08 12:21 v x FALSE
2000-03-10 08:37 o q FALSE
2000-03-11 17:00 v x FALSE

matrix shenanigans

  • David’s scores are based on dyadic winning proportions

 

ID DS normDS
o 2.12 2.42
v 0.45 2.09
q 0.44 2.09
r 0.00 2.00
x -3.02 1.40

Elo-rating

  • sequential processing: winner’s rating increases, loser’s rating decreases
  • deterministic algorithm that requires start ratings and \(k\)
  • magnitude of change depends on \(k\) and the probability of the winner to win
  • classic versions: start ratings and \(k\) are set to fixed values

winning probability you say?

\[ \begin{align} p_{\text{win}} &= \frac{1}{1 - (1 + 10 ^ {(\text{Elo}_A - \text{Elo}_B)/400})}\\ p_{\text{win}} &= \frac{1}{1 + \text{exp}(-0.01(\text{Elo}_B - \text{Elo}_A))}\\ p_{\text{win}} &= \text{normal CDF}(\frac{(\text{Elo}_B - \text{Elo}_A)}{200 \times \sqrt{2}}) \end{align} \]

Elo (1978); Albers and de Vries (2001); Neumann et al. (2011); Franz et al. (2015); Goffe, Fischer, and Sennhenn-Reulen (2018); Sánchez-Tójar, Schroeder, and Farine (2018);

rhetoric questions

  • how would you simulate:
    • a dominance network with steepness 0.86?
    • grooming network with 12% dyads having larger sociality index values than 3.2?
  • if we don’t have a formal underlying model, simulation is impossible (or in itself ad hockery and/or relies on brute force)
  • coming up: one more slide of complaining

more examples of ad hoc solutions in the context of dominance analyses

  • stop because we think we have not observed enough interactions
  • missing dyads? individuals without any observed interactions?
  • what about tied interactions? what about coalitions?
  • observation bias?
  • steepness = regression of standardized scores by ordinal rank
  • randomizing the interaction sequence, really?
  • summing winning probabilities, really?
  • just run a bunch of methods and average?

Sánchez-Tójar, Schroeder, and Farine (2018); Hart et al. (2022)

progress: Elo-rating 2.0

  • set different \(k\) values depending on intensity, set differential start values for groups of animals
  • get ML estimates for start values and \(k\)
Newton-Fisher (2017); Franz et al. (2015)

Elo meets Bayes

  • Bayesian estimation of \(k\), start values and their SD
  • what predicts ratings (age in cueing systems, relatedness)
  • embrace (and use) the uncertainty in rating estimates
Goffe, Fischer, and Sennhenn-Reulen (2018); Neumann and Fischer (2023b)

solution: models with benefits

  • fit principled models, which reflect theory and data generating process

  • Bayesian framework, with all its benefits (and challenges and complexities)

principled models: dominance

principled models: dominance

principled models: dominance

principled models: dominance

principled models: dominance

ad hoc models: bond strength (and other features)

  • SI, CSI, DSI

  • Hinde Index

  • Reciprocity Index

  • SRI, HWI

Silk, Altmann, and Alberts (2006); Silk, Cheney, and Seyfarth (2013)

principled models: bond strength

Hart et al. (2023); Neumann and Fischer (2023a); Ross, McElreath, and Redhead (2024)

uncertainty in parameters

  • this is what proportions of 0 might actually look like (through the Bayesian lens)

  • precision can increase with increasing observation effort

\(Pr(\text{0}) \sim \text{Binomial}([1, 10, 100], \theta)\)

uncertainity in parameters

  • and proportions of 0.5, and it works for rates too

background

steepness measures the degree to which individuals differ from each other in winning dominance encounters

  • incidentally: nothing about ranking in this definition…

  • ‘classic steepness’ is coming from David’s scores

de Vries, Stevens, and Vervaecke (2006)

David’s scores

  • are based on proportions of dyadic winning/losing

David (1987); de Vries, Stevens, and Vervaecke (2006); Neumann and Fischer (2023b)

from DS to steepness

  • propagate uncertainty from DS -> steepness

David (1987); de Vries, Stevens, and Vervaecke (2006); Neumann and Fischer (2023b)

female dominance

the percentage of males dominated by each female (the degree of female dominance)

  • there is nothing about ranking in that definition/operationalization
  • being dominant or not manifests at the level of the dyad
Drews (1993); Kappeler et al. (2022)

a principled model of female dominance

\[ \begin{align} \text{interactions won}_i &\sim \operatorname{Binomial}(\operatorname{logit}^{-1}(\mu_{i}), \text{interactions}_i) \\ \mu_{i} &= b_0 + b_{\text{g}_i} + b_{\text{f}_i} + b_{\text{m}_i} \\ b_{\text{g}_i} &\sim \operatorname{Normal}(0, \sigma_g) \\ b_{\text{f}_i} &\sim \operatorname{Normal}(0, \sigma_f) \\ b_{\text{m}_i} &\sim \operatorname{Normal}(0, \sigma_m) \\ \sigma_g &\sim \operatorname{Exponential}(1) \\ \sigma_f &\sim \operatorname{Exponential}(1) \\ \sigma_m &\sim \operatorname{Exponential}(1) \end{align} \]

a principled model of female dominance

  • it’s a glorified (but principled) way of saying we calculated proportions
  • intercept-only model with varying intercepts for group, female and male
  • actual values (‘BLUPS’) for the varying intercepts for group is the FDI
  • all parameters have priors and posteriors

a principled model of female dominance

 

direct comparison

  • the extremes match fairly well

  • discrepancies in the mid-range

Kappeler et al. (2022)

posterior predictive checks (aka Bayesian glory)

  • wouldn’t it be cool if we could try to predict/simulate interaction outcomes?
  • and compare those predictions to what we actually observed?

model in model

  • how is degree of female dominance associated with interaction features?
  • use posterior of FDI as predictor (not its point estimate)

\[ \begin{align} \text{aggr. act}_i &\sim \operatorname{Bernoulli}(\operatorname{logit}^{-1}(\mu_{i})) \\ \mu_{i} &= \mathcal{A}_i + \mathcal{B}_i * \text{FDI}_i + ... \\ \mathcal{A}_i & = \alpha + \alpha_{\textrm{winner}[i]} + \alpha_{\textrm{loser}[i]} + \alpha_{\textrm{dyad}[i]} + \alpha_{\textrm{group}[i]} & \text{intercept}\\ \mathcal{B}_i & = \beta + \beta_{\textrm{winner}[i]} + \beta_{\textrm{loser}[i]} + \beta_{\textrm{dyad}[i]} + \alpha_{\textrm{group}[i]} & \text{slope for FDI}\\ \text{ints. won}_j &\sim \operatorname{Binomial}(\operatorname{logit}^{-1}(\nu_{j}), \text{ints.}_j) \\ \nu_{j} &= b_0 + \mathbf{\text{FDI}_j} + b_{\text{f}_j} + b_{\text{m}_j} \end{align} \]

female female

  • very similar for male-male and mixed dyads

compare slopes

  • difference in width of posteriors is notable

reproductive success in crested macaque females

  • the association between sociality and reproductive output
  • both grooming and Elo rating are incorporated alongside their measurement error

modeling reproductive success

\[ \begin{align} \text{birth} &\sim \text{Bernoulli}(\text{logit}^{-1}(\mu))\\ \mu &= b_0 + b_1 * \text{E} + b_2 * \text{G}\\ \text{dom. interactions} &\sim f(\text{start ratings}, k)\\ \text{E} &= f(\text{dom. interactions}, \text{start ratings}, k)\\ \text{groom} &\sim \text{Binomial}(\text{tot. point samples}, \text{logit}^{-1}(G))\\ G &\sim \text{Normal}(0, 1) \end{align} \]

  • sample: 456 female-years, 106 females
Duboscq et al. (2023)

uncertainty in social features

  • uncertainty in grooming proportion decreases with more observation effort
  • uncertainty in Elo-rating decreases with more observed interactions

consequences

  • slope in error model is wider (as expected)
  • slope in error model is not pulled toward zero (somewhat surprising)

consequences

  • no change in inference between error model and simpler model without error

wrap up

  • think about the data generating process
  • ask what can I do? what do I want to do?
  • formalize your model as much as possible before fitting it to data
  • uncertainty propagation and dealing with measurement errors is an important aspect of a principled workflow

wrap up

  • we work a lot with proxies of vague theoretical provenance
  • most things we measure, we measure with error
  • but we typically give both these aspects not too much thought

wrap up

  • the systems we study are complex
  • statistics is hard
  • that shouldn’t stop us from trying

thank you

https://xkcd.com/2456

references

Adams, Eldridge S. 2005. “Bayesian Analysis of Linear Dominance Hierarchies.” Animal Behaviour 69: 1191–201. https://doi.org/10.1016/j.anbehav.2004.08.011.
Albers, Paul C H, and Han de Vries. 2001. “Elo-Rating as a Tool in the Sequential Estimation of Dominance Strengths.” Animal Behaviour 61: 489–95. https://doi.org/10.1006/anbe.2000.1571.
Clutton-Brock, Timothy H, Steve D Albon, Robert M Gibson, and Fiona E Guinness. 1979. “The Logical Stag: Adaptive Aspects of Fighting in Red Deer (Cervus Elaphus L.).” Animal Behaviour 27: 211–25. https://doi.org/10.1016/0003-3472(79)90141-6.
David, Herbert A. 1987. “Ranking from Unbalanced Paired-Comparison Data.” Biometrika 74: 432–36. https://doi.org/10.1093/biomet/74.2.432.
de Vries, Han. 1998. “Finding a Dominance Order Most Consistent with a Linear Hierarchy: A New Procedure and Review.” Animal Behaviour 55: 827–43. https://doi.org/10.1006/anbe.1997.0708.
de Vries, Han, Jeroen M G Stevens, and Hilde Vervaecke. 2006. “Measuring and Testing the Steepness of Dominance Hierarchies.” Animal Behaviour 71: 585–92. https://doi.org/10.1016/j.anbehav.2005.05.015.
Douglas, Pamela Heidi, Axel-Cyrille Ngonga Ngomo, and Gottfried Hohmann. 2017. “A Novel Approach for Dominance Assessment in Gregarious Species: ADAGIO.” Animal Behaviour 123: 21–32. https://doi.org/10.1016/j.anbehav.2016.10.014.
Drews, Carlos. 1993. “The Concept and Definition of Dominance in Animal Behaviour.” Behaviour 125: 283–313. https://doi.org/10.1163/156853993X00290.
Duboscq, Julie, Jérôme Micheletta, Dyah Perwitasari-Farajallah, Antje Engelhardt, and Christof Neumann. 2023. “Investigating the Relationship Between Sociality and Reproductive Success in Wild Female Crested Macaques, Macaca Nigra.” International Journal of Primatology 44: 649–69. https://doi.org/10.1007/s10764-023-00376-5.
Elo, Arpad E. 1978. The Rating of Chess Players, Past and Present. New York: Arco.
Franz, Mathias, Emily McLean, Jenny Tung, Jeanne Altmann, and Susan C Alberts. 2015. “Self-Organizing Dominance Hierarchies in a Wild Primate Population.” Proceedings of the Royal Society B: Biological Sciences 282: 20151512. https://doi.org/10.1098/rspb.2015.1512.
Fushing, Hsieh, Michael P McAssey, and Brenda J McCowan. 2011. “Computing a Ranking Network with Confidence Bounds from a Graph-Based Beta Random Field.” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467: 3590–3612. https://doi.org/10.1098/rspa.2011.0268.
Glickman, Mark E. 1999. “Parameter Estimation in Large Dynamic Paired Comparison Experiments.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 48: 377–94. https://doi.org/10.1111/1467-9876.00159.
Goffe, Adeelia S, Julia Fischer, and Holger Sennhenn-Reulen. 2018. “Bayesian Inference and Simulation Approaches Improve the Assessment of Elo-ratings in the Analysis of Social Behaviour.” Methods in Ecology and Evolution 9: 2131–44. https://doi.org/10.1111/2041-210X.13072.
Hart, Jordan D A, Michael N Weiss, Lauren J N Brent, and Daniel W Franks. 2022. “Common Permutation Methods in Animal Social Network Analysis Do Not Control for Non-Independence.” Behavioral Ecology and Sociobiology 76: 151. https://doi.org/10.1007/s00265-022-03254-x.
Hart, Jordan D A, Michael N Weiss, Daniel Franks, and Lauren J N Brent. 2023. “BISoN: A Bayesian Framework for Inference of Social Networks.” Methods in Ecology and Evolution 14: 2411–20.
Kappeler, Peter M., Elise Huchard, Alice Baniel, Charlotte Canteloup, Marie J. E. Charpentier, Leveda Cheng, Eve Davidian, et al. 2022. “Sex and Dominance: How to Assess and Interpret Intersexual Dominance Relationships in Mammalian Societies.” Frontiers in Ecology and Evolution 10: 918773. https://doi.org/10.3389/fevo.2022.918773.
Neumann, Christof, Julie Duboscq, Constance Dubuc, Andri Ginting, Ade Maulana Irwan, Muhammad Agil, Anja Widdig, and Antje Engelhardt. 2011. “Assessing Dominance Hierarchies: Validation and Advantages of Progressive Evaluation with Elo-rating.” Animal Behaviour 82: 911–21. https://doi.org/10.1016/j.anbehav.2011.07.016.
Neumann, Christof, and Julia Fischer. 2023a. “(De)composing Sociality: Disentangling Individual-Specific from Dyad-Specific Propensities to Interact.” https://doi.org/10.1101/2023.08.15.549768.
———. 2023b. “Extending Bayesian Elo-rating to Quantify the Steepness of Dominance Hierarchies.” Methods in Ecology and Evolution 14: 669–82. https://doi.org/10.1111/2041-210X.14021.
Newton-Fisher, Nicholas E. 2017. “Modeling Social Dominance: Elo-ratings, Prior History, and the Intensity of Aggression.” International Journal of Primatology 38: 427–47. https://doi.org/10.1007/s10764-017-9952-2.
Ross, Cody T., Richard McElreath, and Daniel Redhead. 2024. “Modelling Animal Network Data in R Using STRAND.” Journal of Animal Ecology 93 (3): 254–66. https://doi.org/10.1111/1365-2656.14021.
Sánchez-Tójar, Alfredo, Julia Schroeder, and Damien R Farine. 2018. “A Practical Guide for Inferring Reliable Dominance Hierarchies and Estimating Their Uncertainty.” Journal of Animal Ecology 87: 594–608. https://doi.org/10.1111/1365-2656.12776.
Silk, Joan B, Jeanne Altmann, and Susan C Alberts. 2006. “Social Relationships Among Adult Female Baboons (Papio Cynocephalus) I. Variation in the Strength of Social Bonds.” Behavioral Ecology and Sociobiology 61: 183–95. https://doi.org/10.1007/s00265-006-0249-2.
Silk, Joan B, Dorothy L Cheney, and Robert M Seyfarth. 2013. “A Practical Guide to the Study of Social Relationships.” Evolutionary Anthropology 22: 213–25. https://doi.org/10.1002/evan.21367.